Can you suggest some "drills" or "exercises" I can conduct to improve specific parts of my game? I need some some practice and training.

For example, an exercise on setting up an ambush, or taking and holding a point, how to practice taking out tanks, flank an enemy. I dunno, basic tactical drills that every player should be able to recognise, implement, and counter.

2 Answers
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Try some failing strategies first and see if you can figure out why exactly they failed. Watch the replay and try to identify a general mistake on your part (any mistake), and a possible solution. Restart the game, try your new strategy, rinse, repeat :)

For example the taking and holding a point exercise:

start a new skirmish

build as many pioneers/engineers as you can

cap as many points as you can, as far forward as you can

see if you can hold them

Spoiler: you can't, not even against easies. You are basically too overstretched and underdeveloped. Also, your basic units were too weak in comparison to the enemy's.

In your next attempt, maybe build a building first, so you can actually use your resources to build some slightly stronger units, send your pio's/engi's to the relatively close points, send the slightly stronger units more to the front.

In this way you will soon be able to spot and recognize several patterns. Firstly, you'll see the AI is highly predictable. Secondly, as the difficulty rises, they will receive some absurd bonuses. Besides those, you'll recognize that (to stay with the example) an MG is a great way to 'lock down' a point initially.

Build on that knowledge and start playing online. Chances are you'll get plain slaughtered. Keep trying to identify general tactics and counter tactics. You'll get better at spotting them and their variations and countering those.

The biggest thing I did to improve my online abilities was to watch some recordings of some of the better players out there. Pick an army, then work out one or two basic starting strategies - this was enough to get me through quite a few battles. One of the best sites for these IMHO is GameReplays.org

The second thing I'd look at doing is to know exactly what each unit is good at. Did you know, for example, American Infantrymen become more effective the closer they get to the enemy. If you can get them in close combat (i.e. having them very close to another unit) with say Volksgrenadiers, they'll wipe them out in quick fashion. Beware though, they may come out worst against a squad armed with MP40s - submachine guns that excel in close combat. Here's a good article on those Riflemen for you : CoH Wikia

Between those two sites, there's plenty of reading out there to do! Read them up (even write some things down!) then get out there and practice. Good luck soldier! :)